Let $A$ be a primitive row-stochastic matrix. By Perron-Frobenius theorem, $A$ has an eigenvalue 1 and corresponding left eigenvector and right eigenvector $\pi$ and $\mathbb{1}$, i.e., $A\mathbb{1}=1,\pi^TA=\pi$. Let $B=A-\mathbb{1}\pi^T$. I wonder if the 2-norm of $B$ is strictly less than 1?
I don't know how to call $B$, so I just say it's the consensus part of $A$ in the title. I know if $A$ is a doubly-stochastic matrix, the result is true. I have tested several randomly generated row-stochastic matrices, and the result is also true, but I have no idea how to prove it.
Best Answer
Your test was probably flawed. It should be easy to generate a random counterexample, such as $$ A=\frac1{72}\pmatrix{63&9\\ 2&70}, \quad \pi^T=\frac1{11}(2,9), \quad B=A-\mathbf1\pi^T=\frac1{792} \pmatrix{549&-549\\ -122&122}. $$ Numerically we have $\|B\|_2\approx1.00422>1$.